
We note that the norm constraint here has dimension \(c/B\)  and \(\mathbf {w}_{1b}\)  has dimension \(sd/B\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le n/B\min \lbrace sd, c\rbrace \) . We introduce hyperplane arrangements \(\mathbf {D}_{b, j}\)  and enumerate over all of them, yielding
\(p_{BFN}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*(\mathbf {V}_i, \mathbf {Y}_i) \nonumber \\\mathrm {s.t.}~&\max _{\begin{array}{c}\Vert \mathbf {w}_{1b}\Vert _2 \le 1 \\ j \in [P_b] \\ \mathbf {K}_{b, j} \mathbf {w}_{1b} \ge 0\end{array}} \Vert \sum _{i=1}^n {\mathbf {V}_i^{(b)}}^\top \mathbf {D}_{b, j}^{(i)} \mathrm {circ}({\mathbf {X}}_i^{(b)}) \mathbf {w}_{1b}\Vert _2 \le \beta ~\forall b \in [B].\) 
